A228904 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) * y^k ), as read by rows.
1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 26, 62, 1, 1, 5, 70, 1087, 1031, 1, 1, 6, 155, 9257, 124702, 24782, 1, 1, 7, 301, 51397, 4479983, 26375325, 774180, 1, 1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1, 1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1
Offset: 0
Examples
This triangle begins: 1; 1, 1; 1, 2, 1; 1, 3, 7, 1; 1, 4, 26, 62, 1; 1, 5, 70, 1087, 1031, 1; 1, 6, 155, 9257, 124702, 24782, 1; 1, 7, 301, 51397, 4479983, 26375325, 774180, 1; 1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1; 1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1; ... G.f.: A(x,y) = 1 + (1+y)*x + (1+2*y+y^2)*x^2 + (1+3*y+7*y^2+y^3)*x^3 + (1+4*y+26*y^2+62*y^3+y^4)*x^4 + (1+5*y+70*y^2+1087*y^3+1031*y^4+y^5)*x^5 +... The logarithm of the g.f. equals the series: log(A(x,y)) = (1 + y)*x + (1 + 2*y + y^2)*x^2/2 + (1 + 3*y + 15*y^2 + y^3)*x^3/3 + (1 + 4*y + 70*y^2 + 220*y^3 + y^4)*x^4/4 + (1 + 5*y + 210*y^2 + 5005*y^3 + 4845*y^4 + y^5)*x^5/5 + (1 + 6*y + 495*y^2 + 48620*y^3 + 735471*y^4 + 142506*y^5 + y^6)*x^6/6 +... in which the coefficients form A228832(n,k) = binomial(n*k, k^2).
Crossrefs
Programs
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PARI
{T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m*j, j^2)*y^j))+x*O(x^n)), n, x), k, y)} for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))